Extremal statistics for a one-dimensional Brownian motion with a reflective boundary

TL;DR

This paper derives the extremal statistics of a one-dimensional Brownian motion with a reflective boundary, revealing how the maximum displacement and its timing behave over short and long times, with results validated numerically.

Contribution

It provides exact distributions and moments for the maximum displacement and its occurrence time for Brownian motion with a reflective boundary, extending understanding of extremal statistics under boundary constraints.

Findings

The mean maximum displacement scales as rac{cpi}{2}rac{c4t}{c1} for long times.

The distribution of the maximum displacement transitions from free Brownian behavior to boundary-influenced behavior.

The timing of the maximum shifts from symmetric to asymmetric distribution as time increases.

Abstract

We investigate the extreme value statistics of a one-dimensional Brownian motion (with the diffusion constant $D$) during a time interval $\left[0, t \right]$ in the presence of a reflective boundary at the origin, starting from a positive position $x_0$. By deriving the survival probability of the Brownian particle without hitting an absorbing boundary at $x=M$, we obtain the distribution $P(M|x_0,t)$ of the maximum displacement $M$ and its expectation $\langle M \rangle$. In the short-time limit, i.e., $t \ll t_d$ where $t_d=x_0^2/D$ is the diffusion time from the starting position $x_0$ to the reflective boundary at the origin, the particle behaves like a free Brownian motion without any boundaries. In the long-time limit, $t \gg t_d$, $\langle M \rangle$ grows with $t$ as $\langle M \rangle \sim \sqrt{t}$, which is similar to the free Brownian motion, but the prefactor is $\pi/2$ times of the free Brownian motion, embodying the effect of the reflective boundary. By solving the propagator and using a path decomposition technique, we obtain the joint distribution $P(M,t_m|x_0,t)$ of $M$ and the time $t_m$ at which this maximum is achieved, from which the marginal distribution $P(t_m|x_0,t)$ is also obtained. For $t \ll t_d$, $P(t_m|x_0,t)$ looks like a U-shaped attributed to the arcsine law of free Brownian motion. For $t$ equal to or larger than order of magnitude of $t_m$, $P(t_m|x_0,t)$ deviates from the U-shaped distribution and becomes asymmetric with respect to $t/2$. Moreover, we compute the expectation $\langle t_m \rangle$ of $t_m$, and find that $\langle t_m \rangle/t$ is an increasing function of $t$. In two limiting cases, $\langle t_m \rangle/t \to 1/2$ for $t \ll t_d$ and $\langle t_m \rangle/t \to (1+2G)/4 \approx 0.708$ for $t \gg t_d$, where $G\approx0.916$ is the Catalan's constant. All the theoretical results are validated by numerical simulations.